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The inverse problem of finding the time-dependent diffusion coefficient of the heat equation from integral overdetermination data. (English) Zbl 1259.65142

Simultaneous determination of the time-dependent thermal diffusivity and the temperature distribution for one-dimensional heat equation is considered, using the nonlocal boundary and integral overdetermination conditions. The existence and uniqueness of the solution of the inverse problem is shown based on an auxiliary spectral problem and some regularity assumptions of the known data. Result of continuous dependence on the data is also obtained. The inverse problem is then solved by the finite difference method with a predictor-corrector-type approach. Finally, the authors show one numerical example and illustrate the sensitivity of the Crank-Nicolson finite difference scheme combined with an iteration method with respect to noisy over-determination data.

MSC:

65M32 Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs
35K05 Heat equation
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35R30 Inverse problems for PDEs
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